Asymptotic Time Averages and Frequency Distributions

Consider an arbitrary nonnegative deterministic process (in a stochastic setting {X(t), t ≥ 0} is a fixed realization, i.e., sample-path of the underlying stochastic process) with state space S = (−∞,∞). Using a sample-path approach, we give necessary and sufficient conditions for the long-run time average of ameasurable function of process to be equal to the expectation taken with respect to the same measurable function of its long-run frequency distribution. The results are further extended to allow unrestricted parameter (time) space. Examples are provided to show that our condition is not superfluous and that it is weaker than uniform integrability. The case of discrete-time processes is also considered. The relationship to previously known sufficient conditions, usually given in stochastic settings, will also be discussed. Our approach is applied to regenerative processes and an extension of a well-known result is given. For researchers interested in sample-path analysis, our results will give them the choice to work with the time average of a process or its frequency distribution function and go back and forth between the two under a mild condition.


Introduction
In this article we seek weak conditions, necessary and sufficient, for the long-run time average of a process or any measurable function of it to be equal to the expectation taken with respect to its long-run frequency distribution.Throughout the paper we use a sample-path approach (see [1][2][3]) in the sense that we restrict attention to one realization (sample-path) of the process of interest.Our approach reveals that the stochastic assumptions (regenerative, semistationary, etc.) in the most part are not needed for this result to hold, but rather to ensure that the process itself is ergodic and that a stationary distribution exists.In other words, the longrun time averages for different sample-paths of the given stochastic process converge to a common limiting value with probability one (see Example 17 in Section 4).Our approach is intuitive and the proofs are rather elementary; nothing beyond Riemann-Stieltjes integration theory is needed.
Let {(),  ≥ 0} be an arbitrary real-valued rightcontinuous deterministic process (in a stochastic setting {(),  ≥ 0} can be a fixed realization, that is, samplepath of the underlying stochastic process) with state space  = (−∞, ∞).Let where 1{} is an indicator function and ℎ is a real-valued measurable function.It is also assumed that ℎ is integrable with respect to () in the Riemann-Stieltjes sense.Define the following limits when they exist: (2) Let   () and () be the complements of   () and (), respectively (it follows that () = lim →∞   ()).
Note that  and  are the long-run time average of the processes {(),  ≥ 0} and {ℎ(()),  ≥ 0}, respectively, and () is the long-run frequency distribution of {(),  ≥ 0}.In a queueing system  may represent the long-run average number of customers in the system, , and () is the "stationary distribution." At an elementary level the problem can be posed as follows.It is of interest to establish conditions under which the long-run time average of a given process is equal to the expectation taken with respect to its long-run frequency distribution; that is, the following relation holds: It turns out that in our sample-path setting, when −∞ <  < +∞, there is a necessary and sufficient condition for relation (3) to hold.Relation (3) may also be valid even when  = ±∞.
In a stochastic setting relation (3) may have the following interpretation: for each  > 0, let {  (),  ≥ 0} be a process such that, for each ,   () has   () as its distribution function.Then the process {  (),  ≥ 0} represents the status of the original process {(),  ≥ 0} as seen by a random observer that arrives at a random time uniformly distributed between 0 and .If we let   be a random variable with () as its distribution function, then   describes the behavior of the process {  (),  ≥ 0} in steady state, and relation (3) may be written as  =   , where   is the expected value of   .
This problem has theoretical as well as practical significance.For example, in queueing theory the long-run average number of customers in a queueing system is, sometimes, defined as  = lim →∞ ∫  0 ()/, where {(),  ≥ 0} represents the number of customers present in the system (both in queue and service) at instant .However, in applications,  is usually calculated as the expectation of the stationary distribution {  }, of the process {(),  ≥ 0}, provided it exists; that is,  = ∑ ∞ =0   .The question arises as to when the above two quantities are equal, particularly when {(),  ≥ 0} is not stationary and ergodic.Similar questions arise also when calculating other performance measures for queueing systems such as , the long-run average waiting time in the system per customer.Another example arises when dealing with the ASTA (Arrivals that See Time Averages) property (see [1,[4][5][6][7][8]). Following [4,5] the ASTA problem is posed as follows: Given two stochastic processes  ≡ {(),  ≥ 0} and  ≡ {(),  ≥ 0} defined on a common probability space, where  is intended to represent a queue or a network of queues, and  as an arrival point process, let   = inf{ : () ≥ },  ≥ 1, be an imbedded point process, () ⇒ (∞), and (  ) ⇒ X(∞), where ⇒ denotes weak convergence [9].Then the ASTA problem is to find conditions under which (∞)  = X(∞), where  = denotes equality in distribution.However, rather than working to verify (∞)  = X(∞) directly, Melamed and Whitt [4,5] find conditions such that (∞) = (∞), where and ℎ is a bounded measurable real-valued function.In general (∞) and ℎ((∞)) (equivalently (∞) and ℎ( X(∞))) are not necessarily equal unless  is stationary and ergodic.On the other hand, Stidham Jr. and El-Taha [1] prove ASTA by working directly with sample-path versions of (∞) and (∞).In this paper we establish weak conditions (necessary and sufficient) to guarantee the equality of time averages of a process and the expectation taken with respect to its long-run frequency distribution without explicitly assuming stationarity and ergodicity.
In a stochastic framework this problem has been treated by many authors, mostly to establish conditions that guarantee ergodicity.For example, the case of a process with an imbedded stationary sequence (the semistationary process) is proved by [10,11].Franken et al. [12] derive similar results for processes with imbedded marked point processes.Rolski [13] introduces and exploits the notion of ergodically stable processes to prove a similar result.Wolff [14] proves a similar result for processes that are regenerative or nonnegative and stochastically increasing.The case for stochastic clearing processes is considered by [15].Relevant also are [16][17][18][19][20].For references on sample-path analysis, the reader is referred to [1,2,16,[21][22][23][24][25].This article should be of interest to readers with interest in establishing relationships that involve time averages and frequency distributions in a deterministic framework.
El-Taha and Stidham Jr. [2] provide a result in a samplepath setting for the special case of a nonnegative deterministic process.In this article the results of [2] are extended to any function of the process; thus all moments of a process can be treated within one framework.We also extend the state space to allow for the process to take negative as well as positive values.Moreover we consider the case where the parameter space is extended from [0, ∞) to (−∞, ∞).
The article is organized as follows.In Section 2 we concentrate on the case when {(),  ≥ 0} is a general unrestricted process and give necessary and sufficient conditions (see conditions 1 and 2 in Section 2) when a measurable function, satisfying a weak regularity condition, of the process is considered.We also provide some insight into the relationship between our condition and uniform integrability.We also extend our results to allow the parameter  to be unrestricted in sign.Section 3 provides similar results when X is a discrete-time process.In Section 4 three examples are International Journal of Stochastic Analysis 3 given to help clarify the need for condition 1, the relation of condition 1 to uniform integrability, and the relationship between our pure sample-path setting and stochastic settings.In Section 5 we close by looking at regenerative processes.A process with an imbedded sequence is investigated and then used to extend a well-known result for regenerative processes.

Main Results and Related Analysis
In this section we prove the main result, provide several applications, study the connection of the conditions needed to uniform integrability, and point out an extension of the parameter space to allow negative time.Given any deterministic process, we show that for any measurable function of the process the asymptotic time average of a function of a given process is equal to the expectation taken with respect to its asymptotic frequency distribution function under weak conditions.Then several special cases of interest will be stated.Our objective is to seek weak conditions under which the asymptotic time average of a function of a given process is equal to the expectation taken with respect to it asymptotic frequency distribution function; that is, Note that (5) reduces to (3) when ℎ is an identity function.First we establish the following time dependent relationships in Lemmas 1-3, key to proving the main result.Lemma 1.Let ℎ() be any real-valued measurable function and   () a distribution function defined on (−∞, +∞).Then for all  ≥ 0,  > 0 provided all integrals are well defined (i.e., ℎ() is integrable with respect to   () in the Riemann-Stieltjes sense).
Proof.Note that which proves part (i).The proof of (ii) is similar.Part (iii) follows by taking  = 0 in (i) and (ii) and combining both results.
The results in Lemma 1 remain valid if we replace   () by () when the limit exists.Now we give the second partial result.

Lemma 2. For all
Proof.Note that Similarly Then the result follows by appealing to Lemma 1 (iii).
By taking limits, as  → ∞ of both sides of Lemma 2, showing relation (5) holds is seen to be equivalent to a problem of interchanging limits and integrals.The next result is a key lemma.Lemma 3.For all  > 0 and  ≥ 0

International Journal of Stochastic Analysis
Proof.The result will follow if we show that (i) (ii) Now Therefore, using Lemma 1 (i), we obtain which proves part (i).Part (ii) is proved similarly using Lemma 1 (ii).
Because Lemmas 1-3 are time dependent, they can be useful in time dependent analysis of nonstationary stochastic systems.Now we give the main result.Theorem 4. Consider the deterministic process {(),  ≥ 0}, with state space  = (−∞, ∞), and let ℎ(⋅) be any real-valued measurable function.Then, the following are equivalent.
(i) Condition A1: (ii) Condition A2: Proof.Taking the limits as lim →∞ lim →∞ on both sides of Lemma 3 proves the equivalence of (i) and (ii).Now, using Lemmas 3 and 2, we write Suppose 1 (equivalently 2) holds.Now, taking limits as  → ∞ and assuming they exist give Then part (iii) of the theorem follows by taking limits of both sides in ( 22) and ( 23) as  → ∞.Conversely if part (iii) holds, it is straightforward to see that 1 and 2 hold.

Suppose also that condition (i) (equivalently (ii)) of Theorem 4 holds. Then for any measurable real-valued function
Proof.The proof follows by taking limits, as  → ∞, in ( 22) and ( 23) and using a similar argument as in Theorem 4.
So, the result follows by noting that and using Lemma 1 (iii).
In Corollary 6, let ℎ() =   , then (26) gives the th moment of a distribution function which is familiar when () is defined only on [0, ∞).
(ii) From (23), it is clear that if ∫ + − ℎ()() is well defined and finite for all 0 <  < ∞, then  is well defined iff lim →∞  −1 ∫  0 ℎ(())1{() > } is well defined for all  > 0. Now take limits in (23) as  → ∞ to obtain If any two of the three terms in ( 29) are well defined and at least one of them is finite then the third term exists and Corollary 5 holds.
(iii) Assume that all the relevant limits in Corollary 5 are well defined.From (23)   (iv) One can construct sufficient conditions for condition 1 to hold.Following an argument by Billingsley [26, page 186], one can easily see that lim Next we explore the relationship between condition 1 and uniform integrability of the process {  (),  ≥ 0}.
Relation to Uniform Integrability.We discuss the connection between condition 1 and uniform integrability when ℎ is an identity function.Condition 1 of Theorem 4 requires, roughly speaking, that the area, for 0 ≤  ≤ ∞, between  and ()1{() > } minus the area between − and ()1{() < −} goes to zero as  approaches infinity.We point out here that there is a difference between condition 1 and uniform integrability (of   ()) which requires that the above two areas add up to zero as  approaches infinity.Wolff [14] suggests that proving relation (5), in a stochastic setting, is equivalent to showing that the process {  (),  ≥ 0} and   are uniformly integrable (u.i.) in .Condition 1 is weaker than being u.i.; it only coincides with uniform integrability of the process {  (),  ≥ 0} when the process {(),  ≥ 0} is nonnegative.
To shed more light on this difference, we show that the following modified condition is the equivalent to uniform integrability.
Note that in 3 we take the absolute value of ().In our sample-path setting the definition of uniform integrability [26,27] of the process {  (),  ≥ 0} is equivalent to Now, using an argument similar to that used in proving Lemma 3, we obtain Thus condition 3 and (31) are equivalent.We note that condition 3 (equivalently (31)) is sufficient for relation (5) to hold.Example 16 given below shows the existence of a process that obeys relation ( 5), yet condition 3 is not satisfied.

2.1.
Moments.An important special case is when ℎ() =   ,  > 0, that provides a relation between time averages of a process moments and the moments obtained by using its International Journal of Stochastic Analysis asymptotic distribution function.For ℎ() =   , we have the following result.

Extension.
Here, a generalization of Theorem 4 will be considered.We allow the parameter space [0, ∞) to extend to (−∞, ∞) and show that, for any measurable function of the new process, relation (5) remains valid under conditions similar to 1 and 2.Theorem 4 is extended to the case where the parameter can also be negative.
Let {(), −∞ <  < ∞} be an arbitrary (deterministic) process with state space  = (−∞, ∞), and let Here we seek weak conditions under which the following relation holds: Now an argument similar to that of the proof of Theorem 4 will yield this result.
Note that proof of Theorem 9 follows the same lines of argument as that of Theorem 4. Note also that conditions 1 and 2 are similar to conditions 1 and 2, but with 2 in the denominator in 1.

Discrete-Time Processes
In this section we consider a discrete-time process; specifically let {  ,  ≥ 1} be any deterministic discrete-time process, that is, an infinite sequence of real numbers, and let ℎ(⋅) be a real-valued measurable function.Moreover, let and define the following limits when they exist: Similar to the continuous time model, we have the following results.
Theorem 10.Let {  ,  ≥ 1} be any discrete-time process, and let ℎ(⋅) be any a real-valued measurable function.Then, the following are equivalent: Note that when ℎ(⋅) is an identity function, condition (i) of Theorem 10 is sometimes referred to in the literature as the condition for uniform integrability.
Corollary 11.Consider the process {  ,  ≥ 1}.Suppose   () → () uniformly in  as  → ∞.Suppose also that condition (i) (equivalently (ii)) of Theorem 10 holds.Then for any measurable real-valued function This corollary has been found useful in the literature.
by condition (ii) of Theorem 10.The proof of the second statement is similar.
The results in Corollary 12 apply to the continuous time process as well.When the sequence {  ;  ≥ 1} represents, say, service times in queueing model, let ℎ() =  2 in Theorem 10 to, immediately, obtain the following useful results.

Examples and Discussion
In this section, we give three examples.The first example shows that condition 1 is not superfluous.The second example, a modification of the first one, shows that the new modified process does not satisfy the uniform integrability condition (condition 3), yet conditions 1 and 2 are satisfied and therefore relation (5) holds.In the third example, we verify that when condition 1 is satisfied, for a stationary nonergodic stochastic process, even though relation ( 5) is not satisfied in a stochastic setting, it remains valid for every individual sample-path of the process.
, if  = +∞ and condition 1 is not satisfied, we can distinguish to possibilities (a) ∫ ∞ −∞  () = ∞; then relation (5) holds; (b) ∫ ∞ −∞  () < ∞; then relation (5) does not hold and condition 1 takes the value +∞.A similar argument applies when  = −∞.The above discussion should not imply that if condition 1 does not hold it should be infinite.Example 15 given below shows that relation (5) fails with condition 1 assuming a finite value.